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  "body": "\n## Definition\n\nLet $L$ be a [[complete lattice|D7A0ZGZT5S]], and let $\\ll$ be the [[way-below relation|RDM7J5J9HQ]] on $L$.\nThe lattice $L$ is *continuous* if every $y \\in L$ satisfies\n$$y = \\bigvee \\set{x \\in L : x \\ll y},$$\nwhere $\\bigvee$ denotes the [[supremum|0R3H5PANRQ]].\n\n## Remarks\n\nSince $x \\ll y$ implies $x \\le y$, the condition says that $y$ is not larger than the supremum of the elements way below it.\nExample: $L = [0, 1]$ with the usual order, where $x \\ll y$ if and only if $x = 0$ or $x \u003c y$.\nFor $y \u003e 0$ the supremum of $[0, y)$ is $y$, and for $y = 0$ the set $\\set{0}$ has supremum $0$.\nExample: every [[algebraic lattice|3J0BTXSCVN]] is continuous, because each compact $c \\le y$ satisfies $c \\ll y$.\nNon-example: the lattice of closed subsets of $[0, 1]$ under inclusion is complete.\nBut $\\varnothing$ is the only element way below $[0, 1]$, so $[0, 1]$ is not the supremum of the elements way below it.\nA complete lattice is continuous if and only if $\\bigwedge_{j \\in J} \\bigvee D_j = \\bigvee_{f} \\bigwedge_{j \\in J} f(j)$ for every family $(D_j)_{j \\in J}$ of directed subsets, where $f$ runs over the choice functions with $f(j) \\in D_j$.\n",
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